2,882 research outputs found

    Nonlinear convex and concave relaxations for the solutions of parametric ODEs

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    SUMMARY Convex and concave relaxations for the parametric solutions of ordinary differential equations (ODEs) are central to deterministic global optimization methods for nonconvex dynamic optimization and open-loop optimal control problems with control parametrization. Given a general system of ODEs with parameter dependence in the initial conditions and right-hand sides, this work derives sufficient conditions under which an auxiliary system of ODEs describes convex and concave relaxations of the parametric solutions, pointwise in the independent variable. Convergence results for these relaxations are also established. A fully automatable procedure for constructing an appropriate auxiliary system has been developed previously by the authors. Thus, the developments here lead to an efficient, automatic method for computing convex and concave relaxations for the parametric solutions of a very general class nonlinear ODEs. The proposed method is presented in detail for a simple example problem

    Nearly optimal robust secret sharing

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    Abstract: We prove that a known approach to improve Shamir's celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δn, for any constant δ ∈ (0; 1/2). This result holds in the so-called “nonrushing” model in which the n shares are submitted simultaneously for reconstruction. We thus finally obtain a simple, fully explicit, and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k(1+o(1))+O(κ), where k is the secret length and κ is the security parameter. Like Shamir's scheme, in this modified scheme any set of more than δn honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ) while the number of shares n can grow independently. In this case, when n is large enough, the scheme satisfies the “threshold” requirement in an approximate sense; i.e., any set of δn(1 + ρ) honest parties, for arbitrarily small ρ > 0, can efficiently reconstruct the secret

    Optimal rate list decoding via derivative codes

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    The classical family of [n,k]q[n,k]_q Reed-Solomon codes over a field \F_q consist of the evaluations of polynomials f \in \F_q[X] of degree <k< k at nn distinct field elements. In this work, we consider a closely related family of codes, called (order mm) {\em derivative codes} and defined over fields of large characteristic, which consist of the evaluations of ff as well as its first m1m-1 formal derivatives at nn distinct field elements. For large enough mm, we show that these codes can be list-decoded in polynomial time from an error fraction approaching 1R1-R, where R=k/(nm)R=k/(nm) is the rate of the code. This gives an alternate construction to folded Reed-Solomon codes for achieving the optimal trade-off between rate and list error-correction radius. Our decoding algorithm is linear-algebraic, and involves solving a linear system to interpolate a multivariate polynomial, and then solving another structured linear system to retrieve the list of candidate polynomials ff. The algorithm for derivative codes offers some advantages compared to a similar one for folded Reed-Solomon codes in terms of efficient unique decoding in the presence of side information.Comment: 11 page

    On the rates of convergence of simulation based optimization algorithms for optimal stopping problems

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    In this paper we study simulation based optimization algorithms for solving discrete time optimal stopping problems. This type of algorithms became popular among practioneers working in the area of quantitative finance. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in option pricing that illustrates our theoretical findings

    Optimal escape from metastable states driven by non-Gaussian noise

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    5 pages, 2 figures5 pages, 2 figures5 pages, 2 figuresWe investigate escape processes from metastable states that are driven by non-Gaussian noise. Using a path integral approach, we define a weak-noise scaling limit that identifies optimal escape paths as minima of a stochastic action, while retaining the infinite hierarchy of noise cumulants. This enables us to investigate the effect of different noise amplitude distributions. We find generically a reduced effective potential barrier but also fundamental differences, particularly for the limit when the non-Gaussian noise pulses are relatively slow. Here we identify a class of amplitude distributions that can induce a single-jump escape from the potential well. Our results highlight that higher-order noise cumulants crucially influence escape behaviour even in the weak-noise limit

    Technological vs ecological switch and the environmental Kuznets curve

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    We consider an optimal technology adoption AK model in line with Boucekkine Krawczyk and Vallée (2011): an economy, caring about consumption and pollution as well, starts with a given technological regime and may decide to switch at any moment to a cleaner technology at a given permanent or transitory output cost. At the same time, we posit that there exists a pollution threshold above which the assimilation capacity of Nature goes down, featuring a kind of irreversible ecological regime. We study how ecological irreversibility interacts with the ingredients of the latter optimal technological switch problem, with a special attention to induced capital-pollution relationship. We find that if a single technological switch is optimal, one recovers the Environmental Kuznets Curve provided initial pollution is high enough. If exceeding the ecological threshold is optimal, then the latter configuration is far from being the rule.Technology adoption; ecological irreversibility; Environmental Kuznets Curve; Multi-stage optimal control
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